Task $1$: Time Complexity of Nested Loops

$$ Step $1$: Review the time complexity of algorithms involving nested loops.

$$ Step $2$: Analyze the time complexity of the following algorithms:

$$ Algorithm $1$: Sum of elements in a $2$D array $($nested loop$)$

sum $=0$

for i $=0$ to n $-1$ do

for j $=0$ to n $-1$ do

sum $=$ sum $+$ matrix$[$i$][$j$]$

end for

end for

$$ Write the theoretical time complexity using Big$-$O notation.

$$ Algorithm $2$: Printing pairs of elements from two arrays $($nested loop$)$

for i $=0$ to n $-1$ do

for j $=0$ to m $-1$ do

print$($arr$1[$i$],$ arr$2[$j$])$

end for

end for

$$ Analyze the time complexity when n $!=$ m and discuss how the input sizes affect

performance.